Convex eigenfunction of a drifting Laplacian operator and the fundamental gap

Li Ma and Baiyu Liu

Vol. 240 (2009), No. 2, 343–361

DOI: 10.2140/pjm.2009.240.343

Abstract

We study the convexity of the first eigenfunction of the drifting Laplacian operator with zero Dirichlet boundary value provided a suitable assumption to the drifting term is added. We firstly generalize some results of N. Korevaar and S.-T. Yau to gain a Hessian estimate of the first eigenfunction. As an application, we use this Hessian estimate to get a lower bound of the difference of the first and second eigenvalues of the drifting Laplacian. At the end we also find a lower bound when the Hessian estimate does not hold.

Keywords

eigenvalue, drifting Laplacian, Hessian estimate, fundamental gap

Mathematical Subject Classification 2000

Primary: 35P15

Secondary: 35P15

Milestones

Received: 2 June 2008

Revised: 4 June 2008

Accepted: 25 November 2008

Published: 4 March 2009